Let \((N,\xi _N)\)  be a contact manifold in the link of \((M,\xi )\) . We construct a path of length one or two in \(\operatorname{lk}(M,\xi )\)  from \((N,\xi _N)\)  to \((M,\xi )\#(S^3,\xi _1)\) . (We recall that \((M,\xi )\#(S^3,\xi _1)\)  can be obtained from \((M,\xi )\)  by a single contact \((+1)\) -surgery along a Legendrian unknot with \(\operatorname{tb}=-2\)  in a standard Darboux ball in \((M,\xi )\) .)
First, we consider the case that we can obtain \((N,\xi _N)\)  by a contact \((+1)\) -surgery along a Legendrian knot \(K\)  in \((M,\xi )\) . In [1]} it is shown that performing a contact \((+1)\) -surgery \(K\)  followed by a contact \((+1)\) -surgery along a Legendrian meridian \(U\)  of \(K\)  with \(\operatorname{tb}=-1\)  corresponds to a negative stabilization of \((M,\xi )\)  and thus yields \((M,\xi )\#(S^3,\xi _1)\) .
The case that \((N,\xi _N)\)  arises as contact \((-1)\) -surgery from \((M,\xi )\)  can be reduced to the first case by applying Lemma REF  once.
